Time Evolution of Simple Harmonic Oscillation

(1) A = a * sin (ω - k * x) or (2) A = a * sin ( ω + (-k) * x) is the traditional equation to describe relativistic, simple harmonic oscillation.

The author's research into quantum continuity as certainty rather than probability was derived from the observation that revolving objects all have an axis of rotation or spin at the centre origin of revolution. That origin's motion generates angular displacement as a time dependency. It can also be describe as the time evolution of angle. Sine waves describe this perfectly.

The conundrum was than sine waves were of the form (3) A = sin (θ + ϕ),

where k of the dampening constant = -1, and -kx can only be measured in the units of radians/second or rad/s. The units of ω and k * x must match. (3) is a far better representation of the equation at (2), where ω = θ/T and x = θ/t.

Solving the time evolution - the continuum rather than the unit rate - the equation (2) becomes

(4) A = sin ( θ [1] /T + (-θ [2] /t) ) ,

A = sin ( θ [1]/T + (2 * Pi rad + (-θ [2]/t) ) ) ,

A = time evolution of (4) ,

A = sin ( ( θ [1]/T + (2 * Pi rad + (-θ [2]/t) ) ) * t ) ,

A = sin ( θ [1] + θ [2] ) ,

(5) A = sin ( θ [1] + Θ [2] ) The uppercase Θ is used to signify a constant dimension, while θ is a variable dimension. Traditionally Θ is represented by ϕ which is conveniently, visually clearer.

θ would appear to be a constant. The author concluded that in physics every dimension has a constant portion (starting point) and variable portion (the portion which changes with time). From here on, the term for changing with time is respected as time evolution. It is evolved from the product of a rate of unit change and time evolving (expanding). The dependent dimension derived from the product two independent variables of unitary rate and time, as variable period, is the variable dimension, in lowercase while it will be added to a constant dimension in uppercase.

Prudence would place the tradition equation,

A = x + c as

(6) A = C + x , considering that it makes more sense to know the starting point at the beginning, rather than after the variable dimension - the time evolved displacement of the independent unit.

For sine, the ideal way to express the associated equation, eventually, would be

(7) A = sin ( ϕ + θ ).

(8) x = θ

(9) A = a * sin ( θ )

All quantum equations are amplified by a the factor of 1. There is no dampening and no resistance.

Resistance ∝ diameter, where resistance is any kind load.

R = d/2, or a in technology

All of physics is derived from the quantum time evolution equation or the quantum field equation (see the Quantum Field Law article).

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